It is worth noting some very interesting cases when the answer is yes. An amazing [result by Miyata](http://projecteuclid.org/DPubS?service=UI&version=1.0&verb=Display&handle=euclid.kjm/1250524308) states that if $R$ is Noetherian and commutative, $M,N$ are finitely generated  and $E \cong M\oplus N$, any exact sequence
$ 0 \to M \to E \to N \to 0$ must split! 

This holds true slightly more generally, when $R$ is (not necessarity commutative) module-finite over a Noetherian commutative ring. Also, the statement holds for finitely generated pro-finite groups, see Goldstein-Guralnick, J. Group Theory 9 (2006), 317–322.